A carefully sequenced collection of
algebra problem sets
designed to move students from
arithmetic to algebraic problem solving
Kathy Berry
[email protected] OR [email protected]
Independent Mathematics Consultant
Fall 2004
First Steps into Algebra
This collection of problem sets is designed to move students from
their experience with arithmetic into working with variables,
expressions, and equations. It is a starting point for teachers who are
looking for material appropriate for students who have had limited
experience with algebra, or who need a simplified, slower paced
introduction. This material should be appropriate for grade 6 in
particular. Students in grades 5 – 9, or those receiving adjusted
curriculum services, could benefit as well.
Notes to teachers are included for each activity. These provide key
concepts, frequent student errors and misconceptions, and guidance
on using the activities.
Unit 1: From Arithmetic Patterns to Equations
Activity 1:
Activity 2:
Activity 3:
Activity 4:
Activity 5:
Addition and Subtraction Equations – Informally
Multiplication and Division Equations – Informally
Translating everyday language to mathematics
Introducing Variables, Expressions, and Equations
Addition and Subtraction Equations, with
Variables – Informally
Activity 6: Multiplication and Division Equations, with
Variables – Informally
Notes to Teachers on Activities 7 – 13
Activity 7: Solving Addition and Subtraction Equations with
Inverse Operations
Activity 8: Solving Multiplication and Division Equations with
Inverse Operations
Activity 9: Mixed Practice with 1-Step Equations
Activity 10: Mixed Practice with 1-Step Equations – larger numbers
Activity 11: Writing and Solving 1-Step Equations from
Word Problems
Activity 12: Writing Word Problems Modeled by 1-Step Equations
Activity 13: Mixed Practice, 1-Step Equations, Non-Standard Form
Unit 2: Order of Operations, 2-Step Equations
Notes to Teachers for Activities #14 - #17
Activity 14: Working with Exponents (includes negative exponents)
Activity 15: Order of Operations A
Activity 16: Order of Operations B
Activity 17: Recap of Exponents & Order of Operations
Notes to Teachers for Activities #18 - #25
Activity 18: 2-Step Equations - Intuitively
Activity 19: 2-Step Equations – Inverse Operations
Activity 20: 2-Step Equations – Inverse Operations
Activity 21: 2-Step Equations – Word Problems
Activity 22: Mixed Equation Solving (1 & 2 Step) A
Activity 23: Mixed Equation Solving (1 & 2 Step) B
Activity 24: Mixed Word Problems (1 & 2 Step)
Activity 25: Mixed Equation Solving – patterns giving 1, 0
Algebra Activity #1 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve one-step addition and subtraction equations
WITHOUT formal use of inverse operations.
NOTES:
Students have had experience in elementary school with addition and subtraction
fact families:
3+5=8
5+3=8
5=8–3
3=8–5
They are ready to begin extending this toward algebraic representation. A good
first step is to introduce a placeholder – in this case a box or square.
Students should be encouraged to find the missing number. They can find the
number by:
1) inspection (they literally look at the problem and solve it); solving by
inspection requires fluency with fact families and/or with mental mathematics
2) guess and check: students make an initial estimate or guess at the correct
number, check to see if their guess was correct, and then revise their guess in
light of their check. It is not simply guessing until they get it right.
3)informal inverse operations: some students may say “if something plus three
equals eight, then eight minus three equals the missing number”
The purpose of this activity is to encourage students to work with strategies #1 &
#2 above, and move them toward #3 as they get more practice and work with
less convenient numbers.
BOTTOM LINE
Be sure to discuss the starred questions with the students. They
must be encouraged to find patterns in solving these problems, put
those patterns into words, and check to see if they work on new
problems.
Algebra Activity #1
Name_____________
Hour______________
Find the missing values in the problems below:
1. 5 +
=8
2. 9 +
= 15
3. 10 +
= 21
4.
–4=8
5.
– 9 = 17
6.
7.
+ 8 = 23
8.
– 15 = 18
9. 13 +
= 19
10. 21 -
– 13 = 6
=5
11.
+ 12 = 14
12. 30 +
= 55
13.
-15 = 30
14.
- 20 = 60
15. 75 -
= 25
16.
– 50 = 125
17. 90 +
18. 45 -
= 20
19.
– 45 = 20
20.
= 200
-125 = 250
How can you check your work for the problems which have
addition? How can you check your work for the problems which have
subtraction?
Algebra Activity #2– NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve one-step multiplication and division equations
WITHOUT formal use of inverse operations.
NOTES:
Students have had experience in elementary school with multiplication and
division fact families:
5 x 4 = 20
4 x 5 = 20
20 ÷ 4 = 5
20 ÷ 5 = 4
They are ready to begin extending this toward algebraic representation. A good
first step is to introduce a placeholder – in this case a box or square.
Students should be encouraged to find the missing number. They can find the
number by:
1) inspection (they literally look at the problem and solve it); solving by
inspection requires fluency with fact families and/or with mental mathematics
2) guess and check: students make an initial estimate or guess at the correct
number, check to see if their guess was correct, and then revise their guess in
light of their check. It is not simply guessing until they get it right.
3)informal inverse operations: some students may say “if something times 5
equals 20, then 20 divided by 5 equals that something”.
The purpose of this activity is to encourage students to work with strategies #1 &
#2 above, and move them toward #3 as they get more practice and work with
less convenient numbers.
BOTTOM LINE
Be sure to discuss the starred questions with the students. They
must be encouraged to find patterns in solving these problems, put
those patterns into words, and check to see if they work on new
problems.
Algebra Activity #2
Name_____________
Hour______________
Find the missing number in each problem below:
1.
5x
= 20
2.
x 2 = 16
3.
x 3 = 12
4.
÷ 2 = 10
5.
÷5=6
6.
÷4=8
7.
x 7 = 49
8.
x 9 = 63
9.
x1=3
÷ 5 = 20
11.
÷ 2 = 18
12.
÷3=5
=7
14. 56 ÷
=8
15. 90 ÷
10.
13. 28 ÷
16.
19.
x 12 = 144
8x
=0
17.
20.
x 11 = 66 18.
=9
÷ 1 = 25
÷4=0
How can you check your answer to a multiplication problem?
How can you check your answer to a division problem?
Algebra Activity #3 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will generate (with teacher help) a listing of terms related to
the four basic arithmetic operations. This sets the stage for
translating word problems into algebraic representation, and vice
versa.
NOTES:
Students historically have had trouble with word problems (“story problems”) in
mathematics. One way to assist students is to ensure that they understand the
variety of ways the four basic operations can be expressed in everyday
language.
Some words and phrases are listed below. You may also think of others.
Addition: sum, all together, combine, join, unite, increase, total, gain, and
Subtraction: difference, minus, take away, decrease, decreased by, loss, lose,
fewer, less than (as in “5 less than 8 is 3”)
Multiplication: product, of, having so many groups of, twice, thrice, double,
triple, quadruple, rows and columns, array
Division: quotient, fair shares, split up, split among, shared among, shared
between, separate equally (into groups), give out in shares, put into (groups)
equally
You may also want to look at EQUALS: is/are/was/were, results in, comes out
to, gives (“6 times 7 gives 42”), is the same as, the same amount as
BOTTOM LINE
Students need to be able to translate between the English everyday
language and the “foreign language” of mathematics. Students
shouldn’t be tested on knowing the individual words above (other
than the boldface ones), but will need to apply this knowledge in
subsequent activities which require them to write equations and
expressions.
Algebra Activity #3
Name_____________
Hour______________
List as many words or phrases as you can which mean the same
thing as addition, subtraction, multiplication, division, and equality.
Addition
Subtraction
Multiplication
Division
Equal(s)
Algebra Activity #4 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will
• understand the use of variables
• choose variables to represent unknown quantities
• distinguish between equations and expressions
• write multiplication and division expressions using algebraic
conventions (juxtaposition and fraction bars)
NOTES:
This activity is meant to be used as an in-class activity for introducing students to
variables, equations, and expressions. It is NOT intended for an independent
homework assignment.
Variables: Although teachers grew up seeing mainly x’s and y’s in math books,
really any letter can be used for a variable. It is good practice to use a letter
which will remind the problem solver of the quantity for which he or she is
solving.
Did you know… x was used as a variable most of the time because the first
printer of an algebra book had mainly x’s left over to use in the equations after
the rest of the text had been set? Since printing no longer relies on hand-set or
machine-set type, we can use just about any symbol for a variable.
Expressions and Equations:
An expression must contain an operation (add, subtract, multiply, divide, among
others) as well as something to operate on – numbers, variables, or a
combination of these. An algebraic expression has at least one variable.
Otherwise, it is an arithmetic expression.
An equation states that what’s on the left side of the “=” has exactly the same
value as whatever is on the right side of the “=”. Be sure to start identifying with
students the LEFT side and the RIGHT side of the equal sign. You may think
this is trivia, but many students have a hard time visually separating an equation
into LEFT, equal sign, and RIGHT.
Writing Expressions and Equations Using Algebraic Conventions
Multiplication: Writing a number right next to a variable is called juxtaposition.
You can even write more than one variable next to each other, such as abc or
3xy. Really emphasize that when you don’t see an operation sign, but two items
are shoved next to each other, you are supposed to multiply them. This carries
over later into expressions like 5(x-6).
Students may ask, ”Can you write the number after the letter (variable)?”
Well, they could, but since no one else in the world does it that way, their work is
not going to be easily understood. They really should put the number in front of
the variable.
A number being multiplied by a quantity is called a coefficient. In 3xy, 3 is the
numerical coefficient. We would normally just call it the coefficient.
Division: Students MUST get used to seeing division with the fraction bar. When
two or more items are under the bar, it is called a vinculum and acts like a pair of
parentheses as well. Students should read it as division, not as “over”. For
example:
x
= 10 should be read as “x divided by 5 equals 10”, not “x over 5
5
equals 10”.
BOTTOM LINE
Be sure to use the formal vocabulary with students. A helpful way to
do this is to make a statement using less formal vocabulary; then say
exactly the same thing again, but use the formal vocabulary.
Encourage students to use the vocabulary as they work in pairs or
groups, as well as in their writing. Some teachers have found that
having a vocabulary test for each unit is helpful.
Plenty of practice is available on writing expressions from word form.
Check “Algebra With Pizzazz” and “Pre-Algebra With Pizzazz”. Less
practice is available going the other direction – writing words given
expressions or equations.
Algebra Activity #4
Variables, Expressions
and Equations
Name_____________
Hour______________
A. A variable is any symbol used to take the place of a number. It’s
called a variable because its value can change or vary. Normally we
use lower-case letters for variables in algebra. We often use the first
letter of the word of the quantity (value) that we are trying to find.
What would be a good variable for:
1. height
_______
2. width
_______
3. length
_______
4. cost
_______
5. grade
_______
6. years
_______
7. age
_______
8. number of dogs _______
9. number of cats _______
10. some number _______
B. An expression is a two or more numbers or variables or both,
along with an operation (like addition or subtraction, for example).
An equation is a statement that the quantities on each side of the
equal sign are exactly the same value.
Expressions don’t have equal signs. Equations DO.
Write expression or equation next to each item below. Be careful!
11. 3 + 4 = 7 _______________ 12. 3 + n
13. p – 15
_______________
_______________
14. k – 12 = 24 ______________
15. h + 7 – 6 _______________
16. 100 – 15 + 27 + g = 200 __________________
C. We show addition and subtraction with variables the same way
we do when we know both numbers.
EXAMPLE:
3+6=9
x+ 6 = 9
3+x=9
h + 15
r–7
21 + f - 12
24 – 10 = 14
y – 10 = 14
24 – y = 14
Notice that the value of x changes in the example above. So does
the value of y.
We show multiplication and division a little differently when we use
variables. This makes it less confusing because “x” (for multiplying)
and “x” for the variable look too much alike. Raised dots “ • “ also
look too much like decimal points, so we don’t use them either in
algebra. For division, we only use the fraction bar.
WORDS
ARITHMETIC
three times
some number
3 x n or 3 • n
3n
his age times
multiplied by 5
a x 5 or a • 5
5a
10 times 7
10 x 7 or 10•7
10(7) or (10)7
or (10)(7)
the cookies split
c ÷ 5 or 5 c
c
5
n ÷ 8 or 8 n
y
5
into 5 equal groups
some number
divided by 5
ALGEBRA
YOUR TURN: Write an algebraic expression or equation for each
word statement below. Write (X) if it’s an expression, and (Q) if it’s
an equation.
19. some number plus 4
____________________( )
20. some number minus 18
____________________( )
21. Jim’s age increased by 5
____________________( )
22. Tina’s allowance less $4
____________________( )
23. some number times 6
____________________( )
24. some number divided by 8
____________________( )
25. some number plus 8 equals 19
____________________( )
26. 7 multiplied by some number is 21 ____________________( )
27. 5 increased by a number is 8
____________________( )
28. some number split into 2 groups
____________________( )
29. the class separated into 8 groups
____________________( )
30. 40 candies shared among some bags___________________( )
Write a phrase or sentence for each expression or equation below:
31. 20n
32.
y
5
_____________________________________________
_____________________________________________
33. a + 15 = 23 ________________________________________
34. m – 9 = 27
________________________________________
Algebra Activity #5 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve one-step addition and subtraction equations
WITHOUT formal use of inverse operations.
NOTES:
This activity is identical to Activity #1, except the placeholder squares have been
replaced by actual variables. This is the ONLY change between the activities.
Students should be encouraged to find the missing number. They can find the
number by:
1) inspection (they literally look at the problem and solve it); solving by
inspection requires fluency with fact families and/or with mental mathematics
2) guess and check: students make an initial estimate or guess at the correct
number, check to see if their guess was correct, and then revise their guess in
light of their check. It is not simply guessing until they get it right.
3)informal inverse operations: some students may say “if something plus three
equals eight, then eight minus three equals the missing number”
The purpose of this activity is to get students used to seeing variables in
equations. Encourage students to work with strategies #1 & #2 above, and move
them toward #3 as they get more practice and work with less convenient
numbers.
Students should be encouraged to write their answer as: a = 3, and either circle
or box their answer. This begins good habits they will use as they progress to
more challenging equations. Organizing work is a key skill in learning algebra.
BOTTOM LINE
Be sure to discuss the starred questions with the students. They
must be encouraged to find patterns in solving these problems, put
those patterns into words, and check to see if they work on new
problems.
Algebra Activity #5
Name_____________
Hour______________
Find the value for each variable in the problems below:
1.
5 + a
4.
e –
7.
h
=8
4=8
+ 8 = 23
10. 21 - m
=5
13.
r - 15 = 30
16.
19.
2. 9 + c
= 15
3. 10 + d
= 21
5.
f – 9 = 17
6.
8.
j – 15 = 18
9. 13 + k
= 19
12. 30 + p
= 55
11.
n
+ 12 = 14
g – 13 = 6
14. s - 20 = 60
15. 75 -
t
= 25
u – 50 = 125
17. 90 + v
18. 45 - w
= 20
x – 45 = 20
20.
= 200
y -125 = 250
How can you check your work for the problems which have
addition? How can you check your work for the problems which have
subtraction?
Algebra Activity #6 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve one-step multiplication and division equations
WITHOUT formal use of inverse operations.
NOTES:
This activity is identical to Activity #1, except the placeholder squares have been
replaced by actual variables. This is the ONLY change between the activities.
Students should be encouraged to find the missing number. They can find the
number by:
1) inspection (they literally look at the problem and solve it); solving by
inspection requires fluency with fact families and/or with mental mathematics
2) guess and check: students make an initial estimate or guess at the correct
number, check to see if their guess was correct, and then revise their guess in
light of their check. It is not simply guessing until they get it right.
3) informal inverse operations: some students may say “if something times 5 is
20, then 20 ÷ 5 is that number”.
The purpose of this activity is to get students used to seeing variables in
equations. Encourage students to work with strategies #1 & #2 above, and move
them toward #3 as they get more practice and work with less convenient
numbers.
Students should be encouraged to write their answer as: a = 4, and either circle
or box their answer. This begins good habits they will use as they progress to
more challenging equations. Organizing work is a key skill in learning algebra.
BOTTOM LINE
Be sure to discuss the starred questions with the students. They
must be encouraged to find patterns in solving these problems, put
those patterns into words, and check to see if they work on new
problems.
Algebra Activity #6
Name_____________
Hour______________
Find the value of the variable in each problem below:
1.
5a
4.
d
2
7.
10.
= 20
= 10
7g = 49
k
= 20
5
28
13.
p
=7
2.
2b = 16
3.
3c = 12
5.
e
3
6.
f
4
8.
9h = 63
9.
10j = 30
m
2
11.
14.
= 18
56
=8
q
16.
12s = 144
17.
11t
= 66
19.
8v
20.
w
4
=0
=0
=6
=8
12.
n
3
15.
90
=9
r
18.
u
1
=5
= 25
How can you check your answer to a multiplication problem?
How can you check your answer to a division problem?
Algebra Activities #7 – 13
NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
In Activities 7 – 13, students begin solving 1-step equations using
inverse operations. They also solve and write word problems which
can be modeled by 1-step equations.
NOTES:
When presenting inverse operations to the students, an easy way to talk about
them is as one of the “Golden Rules of Algebra”. Most students are familiar with
a variation of this…
“In algebra, what you do to one side of the equal sign, you MUST do to the other
side.”
Algebra is very fair and equitable in this way. That seems to appeal to children.
You will really need to hammer home by the end of this unit that:
addition is the opposite (inverse) of subtraction
subtraction is the opposite (inverse) of addition
multiplication is the opposite (inverse) of division
division is the opposite (inverse) of multiplication
Please do use the I-word (inverse) right along with the less formal word,
“opposite”.
You will also need to show students the many ways we show multiplication. It
may confuse some students as to why there are so many ways to write
multiplication. It may help to explain that mathematics has evolved and changed
over time, and it is doing so today! As printing improved, and more people
learned about algebra, different symbols were tried out – some remain with us
today, and some were discarded as too hard or confusing to use.
BOTTOM LINE: Students need to learn to solve AND check their
work in an organized fashion. Using inverse operations is the first
formal step into algebra.
Algebra Activity #7 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve one-step addition and subtraction equations WITH
formal use of inverse operations.
NOTES:
Prior to this activity, students have solved by inspection, by guess and check,
and by informal observation of patterns in equation solving. Now it’s time to
begin moving them to using inverse operations.
The hardest part at this point is to get students to show their work in a consistent,
organized way. Many students can solve by inspection, and see no reason to
show work. They need to know that they have to practice showing work and
following format on easy problems, so that when they get to harder ones, the
format they use can help them move from step to step.
Young students typically show their equation solving in this way:
x + 16 = 36
- 16 -16
x = 20
?
CK: 20 + 16 = 36
36 = 36 √
Students should be encouraged to write their
answer as: x=20, and either circle or box their
answer. This begins good habits they will use as
they progress to more challenging equations.
Organizing work is a key skill in learning algebra.
Students should also be REQUIRED to check their
work every time a new skill or strategy is
introduced.
BOTTOM LINE
Hang in there when students complain about all the writing. Algebra
is NOT a difficult subject; two of the keys are following directions and
staying organized. You may wish to give points for following format
as well as for finding the correct answer.
Algebra Activity #7
Name_____________
Hour______________
Solve the equations below by using inverse operations. Show your
work. Then, check your work and show your check.
1. x + 16 = 36
2. p + 5 = 18
3. t + 10 = 22
CK:
CK:
CK:
4. d – 14 = 20
5. v – 8 = 16
6. r – 21 = 31
CK:
CK:
CK:
7. 12 + h = 23
8. 14 + w = 25
9. 15 + n = 65
CK:
CK:
CK:
10. g + 24 = 38
11. b – 9 = 90
12. x – 43 = 57
CK:
CK:
CK:
Algebra Activity #8 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve one-step multiplication and division equations
WITH formal use of inverse operations.
NOTES:
Prior to this activity, students have solved by inspection, by guess and check,
and by informal observation of patterns in equation solving. Now it’s time to
begin moving them to using inverse operations.
The hardest part at this point is to get students to show their work in a consistent,
organized way. Many students can solve by inspection, and see no reason to
show work. They need to know that they have to practice showing work and
following format on easy problems, so that when they get to harder ones, the
format they use can help them move from step to step.
Young students should show their equation solving in this way:
MULTIPLICATION:
5a = 20
We generally don’t rewrite every step on it’s own line like
this. It is shown here for clarity. We typically draw a
fraction bar under both sides of the original problem, and
place the 5 underneath. Then, we cancel the 5s without
rewriting the problem. The only “new” line we write is the
answer, followed by the check.
5a 20
=
5
5
5a 20
=
5
5
4
a=4
?
CK: 5(4) = 20
20 = 20 √
Students should be encouraged to write their answer as:
a = 4, and either circle or box their answer. This begins
good habits they will use as they progress to more
challenging equations. Organizing work is a key skill in
learning algebra. Students should also be REQUIRED to
check their work every time a new skill or strategy is
introduced.
Division is illustrated on the next page.
Division Equations
Students should show their work in this way:
m
= 12
6
(6 )
(6 )
m
= 12(6 )
6
Note use of ( ) to show multiplication.
m
= 12(6 )
6
Cancel the 6s, which leaves a factor of “1”.
m = 72
CK:
DO NOT use x’s or raised dots.
1 times m is just m.
Final answer? Better check. Many students will write “2”
as the answer for this problem.
72 ?
= 12
6
12 = 12√
Go back and circle your answer since it’s correct.
m = 72
Again, we actually don’t rewrite each step on a separate line. It is shown here for
clarity.
BOTTOM LINE
Hang in there when students complain about all the writing. Algebra
is NOT a difficult subject; two of the keys are following directions and
staying organized. You may wish to give points for following format
as well as for finding the correct answer.
Algebra Activity #8
Name_____________
Hour______________
Solve the equations below by using inverse operations. Show your
work. Then, check your work and show your check.
1. 4n = 12
2. 5h = 35
3. 10g = 60
CK:
CK:
CK:
4.
m
= 12
6
CK:
7.
5.
k
= 20
4
CK:
12d = 72
8.
CK:
CK:
10. 20x = 160
11.
CK:
CK:
6.
x
= 12
3
CK:
c
=3
9
9.
w
=4
8
CK:
x
= 40
8
12. 100m = 4000
CK:
Algebra Activity #9 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve one-step equations WITH formal use of inverse
operations. Problems are mixed with addition, subtraction,
multiplication, and division.
This activity gives students the opportunity to mentally shift among
the four operations and their inverses.
Students should show work using the proper format for each type of
problem. Students should also check their work for each problem.
COMMON QUESTIONS AND MISCONCEPTIONS
Students may ask if they can write “12 = a” rather than “a = 12”.
Technically speaking, they are the same thing, and one is not more correct than
the other. They will generally find, though, that most people write “variable =
answer”, since their job was to find the value of the variable.
Division problems in particular cause students headaches. They
often follow this route of logic…
w
=8
4
Hmmm, I see a division problem with an 8 and a 4. So, 8 ÷ 4 = 2.
Wow, that was easy, let’s go on to the next one!
If students are required to check their work, this gets cleared up quickly! They
easily see that
2
= 8 is NOT a true statement. We need to work to have
4
students get to the point where they can say “in a problem like this, just take the
denominator times the whole number, then check”. Students need lots of
practice opportunities to get to this point; don’t tell them this shortcut too soon, or
it just becomes another rule that they don’t understand and often apply at
inappropriate times.
BOTTOM LINE: Following format and checking your work pays off!
Algebra Activity #9
Name_____________
Hour______________
Solve the equations below by using inverse operations. Show your
work. Then, check your work and show your check.
x
= 14
2
1. 4n = 32
2. 5h = 15
3.
CK:
CK:
CK:
4.
m
=3
2
5.
k + 19 = 25
6. y – 17 = 20
CK:
CK:
CK:
7. 12 + d = 72
8.
CK:
CK:
CK:
10. x - 20 = 160
11. f – 16 = 36
12. 9c = 36
CK:
CK:
CK:
c
= 10
9
9.
w
=8
4
Algebra Activity #10 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve one-step equations WITH formal use of inverse
operations. Problems are mixed with addition, subtraction,
multiplication, and division. This activity uses larger numbers than
#9.
This activity gives students the opportunity to mentally shift among the four
operations and their inverses. Students should feel more pressed to use inverse
operations as the numbers are larger.
Students should show work using the proper format for each type of problem.
Students should also check their work for each problem.
COMMON QUESTIONS AND MISCONCEPTIONS
Students may ask if they can write “12 = a” rather than “a = 12”.
Technically speaking, they are the same thing, and one is not more correct than
the other. They will generally find, though, that most people write “variable =
answer”, since their job was to find the value of the variable.
Division problems in particular cause students headaches. They
often follow this route of logic…
w
=8
4
Hmmm, I see a division problem with an 8 and a 4. So, 8 ÷ 4 = 2.
Wow, that was easy, let’s go on to the next one!
If students are required to check their work, this gets cleared up quickly! They
easily see that
2
= 8 is NOT a true statement. We need to work to have
4
students get to the point where they can say “in a problem like this, just take the
denominator times the whole number, then check”. Students need lots of
practice opportunities to get to this point; don’t tell them this shortcut too soon, or
it just becomes another rule that they don’t understand and often apply at
inappropriate times.
BOTTOM LINE: Following format and checking your work pays off!
Algebra Activity #10
Mixed Equation Solving Practice
Name_____________
Hour______________
Solve the equations below by using inverse operations. Show your
work. Then, check your work and show your check.
1. 4n = 72
2. 5h = 150
CK:
CK:
4.
m
=3
12
5.
3.
x
= 24
2
CK:
k + 39 = 75
6. y – 170 = 200
CK:
CK:
CK:
7. 42 + d = 142
8.
CK:
CK:
CK:
10. x - 45 = 160
11. f – 116 = 354
12. 9c = 360
CK:
CK:
CK:
c
= 15
9
9.
w
=2
14
Algebra Activity #11 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will solve real-world problems which can be modeled with
one-step equations. Students are expected to solve their equations
WITH formal use of inverse operations. Problems are mixed with
addition, subtraction, multiplication, and division.
This activity gives students the opportunity to represent real-life
situations with algebraic equations, and to solve those equations.
Students should show work using the proper format for each type of
problem. Students should also check their work for each problem.
Their answer MUST make sense in the context of the problem.
COMMON QUESTIONS AND MISCONCEPTIONS
For #1, the teacher writes: 8 + p = 21 or p + 8 = 21. Some students
wrote 21 – 8 = p. Are the students wrong?
No – the students did write an algebraic equation. It does contain a variable.
The students’ equation, though, will not require an inverse operation to solve it.
Teachers will want to take advantage of situations like this to show how each of
the three representations above is equivalent, and all lead to the same result.
Should I require students to answer in a complete sentence, or
is “p = 13” enough?
Writing “p = 13” is the solution to an algebraic equation, and doesn’t make sense
written that way for the original problem. The original problem asks about Pat’s
guests. Students should answer the question that was asked…
“Pat had invited 13 guests.” The students will whine and moan about the
writing, but they’ll get over it; just shrug it off and continue requiring a complete
sentence answer.
BOTTOM LINE: Setting up and solving word problems takes A LOT of time
and A LOT of experience to do well. It’s not all going to happen right now.
Algebra Activity #11
Solving Real-World Problems
Name_____________
Hour______________
For each problem below:
1) define your variable (what are you trying to find?)
2) write an equation
3) solve your equation
4) check your work
5) answer the question in the problem
6) make sure your answer makes sense in the problem situation
1. Chris invited 8 guests to a party. Pat also invited several people
to the same party. If there were 21 people at the party, how many
had Pat invited?
2. Lisa went to the mall and bought a new CD for $12. She then
went to Taco Bell and noticed she had only $3 in her wallet. How
much money did she have before buying the CD?
3. Terry has 5 dogs. He bought them 40 lbs of Delicious Dog Food.
If each dog gets the same amount, how much does each dog get?
4. Andrea’s uncle died, and the value of his estate was shared
between Andrea and her three sisters. If each sister received
$30,000, what was the value of the whole estate?
5. The school sold 3,141 “pie in the face” ticket this year. This is 219
more than was sold last year. How many tickets were sold last year?
6. Erica is building a new foundation for a house. Each concrete
truck carries 9 cubic yards of concrete. She needs 36 cubic yards
for this job. How many truckloads will be needed?
Algebra Activity #12 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students will write a real-world problem given several one-step
algebraic equations. They then will solve and check the equation.
NOTES:
Students are often asked to take a word problem and model it with an
equation. They are rarely given the opportunity to go the other way.
You may want to review Activity #3 which lists a number of ways of
talking about the four basic operations and equals.
As a suggestion, have students share with the class the problems
they create. Perhaps have four students write their word problem for
number 1 on transparencies. Then, as a class decide why each of
them can be modeled by “x + 4 = 7”.
BOTTOM LINE:
Students need the opportunity to move between representations of
problem situations. It takes a while to develop proficiency in working
with word problems. Don’t expect proficiency yet.
Algebra Activity #12
Name_____________
Hour______________
For each equation below, write a word problem. Then solve the equation and
see if your answer makes sense (check it).
1.
x+4=7
2.
c–3=8
3.
3r = 18
4.
g
=3
5
5.
4n = 24
6.
w
=5
10
Algebra Activity #13 – NOTES TO TEACHERS
GRADE RANGE: 5 – 9
DESCRIPTION:
Students have an opportunity to practice solving equations using
inverse operations. Some of the equations are written with the
constant (the “equals number”) first.
NOTES:
By this point, students are probably pretty comfortable solving
equations in the following forms:
x+b=c
x–b=c
ax = c
x
=c
a
Many students are baffled when the constant (c) is on the LEFT side
of the equal sign and the variable is on the right. They need to see
many such examples and be pushed to accept that when quantities
are equal, it really doesn’t matter which side comes first.
Some students insist upon rewriting the original equation so that the
constant is to the right. You may choose to allow this, but don’t let
them say they’re rewriting it the “right” way, as in the only correct
way.
BOTTOM LINE:
You’ll start noticing some students being very proficient at equation
solving and being ready to start taking shortcuts. Other students
remain committed to not showing work and thus making careless
mistakes. You may need to split the class into small groups and
allow those ready to do the work in their head to do so, while others
get individual attention to do careful and accurate work.
Algebra Activity #13
Mixed Practice – 1 Step Equations
Name_____________
Hour______________
Solve each equation below. Be sure to check your work.
1.
12h = 36
2. 50 = 25t
3. 18 = r + 12
4.
h – 27 = 50
5. 28 = y – 14
6. 10 =
8. 100 = c – 30
9. 50 = 43 + j
12. 70r = 630
7.
k
=7
11
10.
125 = 5n
11.
13.
266 = 7x
14. 337 = p + 337
16. p – 60 = 83
17.
875 = 225 + x
f
= 33
10
g
12
15. 17m = 0
18.
45 =
a
9